This worked example shows:
Here we analyze population-level data of the wildflower Dianthus carthusianorum (common name: Carthusian pink) in 65 calcareous grassland patches in the Franconian Jura, Germany (Rico et al. 2013):
All required packages should have been installed already when you installed ‘LandGenCourse’.
Note: the function ‘library’ will always load the package, even if it is already loaded, whereas ‘require’ will only load it if it is not yet loaded. Either will work.
library(LandGenCourse)
#require(here)
#require(spdep)
#require(nlme)
#require(lattice)
#require(MuMIn)
library(spatialreg)
library(ggplot2)
library(ggmap)
source(system.file("extdata", "panel.cor.r",
package = "LandGenCourse"))Package ‘spmoran’ not automatically installed with ‘LandGenCourse’:
We will model allelic richness ‘A’ as a function of the following predictors:
The connectivity indices Si were calculated for each focal patch i, integrating over all other patches j where the species was present (potential source patches) using Hanski’s incidence function.
Allelic richness ‘A’ was not calculate for populations with < 5 individuals. Here we extract only the patches with ‘A’ values, and the variables needed, and store them in a data frame ‘Dianthus.df’.
The functions in package ‘ggmap’ expect the projection to be Longitude/Latitude. We could start by transforming the projection of the Dianthus dataset:
Dianthus.longlat <- sp::spTransform(Dianthus,
CRSobj = tmaptools::get_proj4("longlat")$proj4string)
make_bbox(lon = Longitude, lat = Latitude, data = Dianthus@data, f = 0.2)## left bottom right top
## 10.91051 48.90282 11.08951 49.07073
Here, however, we use a shortcut as the ‘Dianthus@data’ dataframe already contains variables with Longitude and Latitude. We use these to define the bounding box for the map, extending the map extent by 20% (‘f=0.2’).
Now we can obtain a stamen map for the study area from the internet and add points for the sampling sites. Here we use a terrain map.
Note: here we use the ‘terrain’ map from source ‘stamen’, which is freely available. Starting in mid 2018, accessing maps from Google with ‘source = google’ requires an API key.
ggmap(get_stamenmap(bbox, maptype = "terrain", zoom=12)) +
geom_point(aes(x = Longitude, y = Latitude), data = Dianthus.longlat@data,
colour = "black", size = 2)## Source : http://tile.stamen.com/terrain/12/2172/1408.png
## Source : http://tile.stamen.com/terrain/12/2173/1408.png
## Source : http://tile.stamen.com/terrain/12/2174/1408.png
As you can see from the map, most sites lie on the steep slopes between an upper and a lower Jurassic plateau. A few sites lie at the forest edge on the upper plateau, typically in areas where the soil is too shallow to allow crop farming. With in the study area, all known sites were sampled. Additional sites are expected to be found mainly in the valley system in the Southwest.
There are three types of stamen maps: terrain, toner (black and white), and watercolor. Terrain and toner maps have three components each (background, labels, lines), which can be downloaded individually if needed. Hence, the argument ‘maptype’ has the following options:
Let’s see what a ‘toner’ map would look like:
ggmap::ggmap(get_stamenmap(bbox, maptype = "toner", zoom=12, force = TRUE)) +
geom_point(aes(x = Longitude, y = Latitude), data = Dianthus.longlat@data,
colour = "black", size = 2) When fitting linear models, it is always a good idea to look at the correlations first.
Dianthus.df <- data.frame(A=Dianthus@data$A, IBD=Dianthus@data$Eu_pj,
IBR=Dianthus@data$Sheint_pj,
PatchSize=log(Dianthus@data$Ha),
Longitude=Dianthus@data$Longitude,
Latitude=Dianthus@data$Latitude,
x=Dianthus@coords[,1], y=Dianthus@coords[,2])
Dianthus.df <- Dianthus.df[!is.na(Dianthus.df$A),]
dim(Dianthus.df)## [1] 59 8
Questions:
Let’s also check the association between patch size and population size:
boxplot(log(Dianthus$Ha) ~ Dianthus$pop09, ylab="PatchSize (log(Ha))",
xlab="Population size category")Even though the population size categories were very broad, there appears to be a strong relationship between populations size (category) and (the logarithm of) patch size.
Despite this relationship, connectivity models Si that only considered Dianthus carthusianorum presence/absence (‘pj’) in source patches ‘j’ were better supported than those Si models that took into account source patch area (‘Aj’) or population size (‘Nj’).
We can check this by calculating the correlation of allelelic richness ‘A’ with each of the 15 connectivity models ‘Si’ in the data set.
round(matrix(cor(Dianthus@data$A, Dianthus@data[,15:29],
use="pairwise.complete.obs"), 5, 3, byrow=TRUE,
dimnames=list(c("Eu", "Shecte", "Sheint", "Shenu", "Forest"),
c("pj", "Aj", "Nj"))),3)## pj Aj Nj
## Eu 0.024 0.207 0.098
## Shecte 0.375 0.273 0.369
## Sheint 0.403 0.196 0.369
## Shenu 0.135 -0.045 0.105
## Forest 0.137 0.168 0.138
Here we fit three multiple regression models to explain variation in allelic richness:
##
## Call:
## lm(formula = A ~ IBD, data = Dianthus.df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.68545 -0.10220 0.03883 0.16178 0.36100
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.070960 0.064061 63.549 <2e-16 ***
## IBD 0.008454 0.047460 0.178 0.859
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2324 on 57 degrees of freedom
## Multiple R-squared: 0.0005563, Adjusted R-squared: -0.01698
## F-statistic: 0.03173 on 1 and 57 DF, p-value: 0.8593
This model does not fit the data at all!
##
## Call:
## lm(formula = A ~ IBR, data = Dianthus.df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.66844 -0.11251 0.03418 0.12219 0.41760
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.92306 0.05499 71.348 < 2e-16 ***
## IBR 0.25515 0.07672 3.326 0.00155 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2128 on 57 degrees of freedom
## Multiple R-squared: 0.1625, Adjusted R-squared: 0.1478
## F-statistic: 11.06 on 1 and 57 DF, p-value: 0.001547
This model fits much better. Let’s check the residuals plots.
The residuals show some deviation from a normal distribution. Specifically, the lowest values are lower than expected.
##
## Call:
## lm(formula = A ~ PatchSize + IBR, data = Dianthus.df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.68359 -0.08844 0.02538 0.11705 0.41705
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.05158 0.07781 52.069 <2e-16 ***
## PatchSize 0.04266 0.01888 2.260 0.0277 *
## IBR 0.11338 0.09709 1.168 0.2479
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2055 on 56 degrees of freedom
## Multiple R-squared: 0.2325, Adjusted R-squared: 0.2051
## F-statistic: 8.482 on 2 and 56 DF, p-value: 0.0006058
This combinde model explains more variation in allelic richness than the IBR model alone. Moreover, after adding PatchSizes, the IBR term is no longer statistically significant!
Has the distribution of residuals improved as well?
Not really!
Before we interpret the models, let’s check whether the assumption of independent residuals is violated by spatial autocorrelation in the residuals.
To calculate and test Moran’s I, we first need to define neighbours and spatial weights. Here we use a Gabriel graph to define neighbours.
We define weights in three ways (see Week 5 video and tutorial for code explanation):
In each case, we row-standardize the weights with the option ‘style = “W”’.
Note: when using ‘graph2nb’, make sure to use the argument ‘sym=TRUE’. This means that if A is a neighbour of B, B is also a neighbour of A. The default is ‘sym=FALSE’, which may result in some sites not having any neighbours assigned (though this would not be evident from the figure!).
xy <- data.matrix(Dianthus.df[,c("x", "y")])
nb.gab <- spdep::graph2nb(spdep::gabrielneigh(xy), sym=TRUE)
par(mar=c(0,0,0,0))
plot(nb.gab, xy)listw.gab <- spdep::nb2listw(nb.gab)
dlist <- spdep::nbdists(nb.gab, xy)
dlist <- lapply(dlist, function(x) 1/x)
listw.d1 <- spdep::nb2listw(nb.gab, style = "W", glist=dlist)
dlist <- lapply(dlist, function(x) 1/x^2)
listw.d2 <- spdep::nb2listw(nb.gab, style = "W", glist=dlist)Now we can quantify and test Moran’s I for each variable to test for spatial autocorrelation in response and predictor variables. For now, we’ll take the simple weights ‘listw.gab’.
##
## Moran I test under randomisation
##
## data: Dianthus.df$A
## weights: listw.gab
##
## Moran I statistic standard deviate = 2.881, p-value = 0.001982
## alternative hypothesis: greater
## sample estimates:
## Moran I statistic Expectation Variance
## 0.29180913 -0.01724138 0.01150691
##
## Moran I test under randomisation
##
## data: Dianthus.df$IBD
## weights: listw.gab
##
## Moran I statistic standard deviate = 5.9689, p-value = 1.194e-09
## alternative hypothesis: greater
## sample estimates:
## Moran I statistic Expectation Variance
## 0.62709841 -0.01724138 0.01165320
##
## Moran I test under randomisation
##
## data: Dianthus.df$IBR
## weights: listw.gab
##
## Moran I statistic standard deviate = 3.008, p-value = 0.001315
## alternative hypothesis: greater
## sample estimates:
## Moran I statistic Expectation Variance
## 0.30918167 -0.01724138 0.01177588
##
## Moran I test under randomisation
##
## data: Dianthus.df$PatchSize
## weights: listw.gab
##
## Moran I statistic standard deviate = 3.4257, p-value = 0.0003066
## alternative hypothesis: greater
## sample estimates:
## Moran I statistic Expectation Variance
## 0.35490160 -0.01724138 0.01180103
Questions:
Next, let’s test each model for autocorrelation in the residuals:
##
## Global Moran I for regression residuals
##
## data:
## model: lm(formula = A ~ IBD, data = Dianthus.df)
## weights: listw.gab
##
## Moran I statistic standard deviate = 2.9439, p-value = 0.00162
## alternative hypothesis: greater
## sample estimates:
## Observed Moran I Expectation Variance
## 0.28889983 -0.02854559 0.01162751
##
## Global Moran I for regression residuals
##
## data:
## model: lm(formula = A ~ IBR, data = Dianthus.df)
## weights: listw.gab
##
## Moran I statistic standard deviate = 1.883, p-value = 0.02985
## alternative hypothesis: greater
## sample estimates:
## Observed Moran I Expectation Variance
## 0.18032443 -0.02296810 0.01165614
##
## Global Moran I for regression residuals
##
## data:
## model: lm(formula = A ~ PatchSize + IBR, data = Dianthus.df)
## weights: listw.gab
##
## Moran I statistic standard deviate = 1.7484, p-value = 0.04019
## alternative hypothesis: greater
## sample estimates:
## Observed Moran I Expectation Variance
## 0.16223745 -0.02708922 0.01172530
Quite a bit of the spatial autocorrelation in allelic richness can be explained by the spatial structure in the predictors IBR and PatchSize. There is still statistically significant spatial autocorrelation in the residuals, though it is not strong any more.
An alternative way to account for spatial autocorrelation in the residuals is spatial regression with a simultaneous autoregressive error model (SAR).
The method ‘errorsarlm’ fits a simultaneous autoregressive model (‘sar’) to the error (‘error’) term of a ‘lm’ model.
This approach is based on spatial neighbours and weights. We have already defined them in three versions of a ‘listw’ object. Let’s see which one fits the data best.
mod.sar.IBR.gab <- spatialreg::errorsarlm(A ~ PatchSize + IBR, data = Dianthus.df,
listw = listw.gab)
mod.sar.IBR.d1 <- spatialreg::errorsarlm(A ~ PatchSize + IBR, data = Dianthus.df,
listw = listw.d1)
mod.sar.IBR.d2 <- spatialreg::errorsarlm(A ~ PatchSize + IBR, data = Dianthus.df,
listw = listw.d2)
MuMIn::model.sel(mod.lm.IBR, mod.sar.IBR.gab, mod.sar.IBR.d1, mod.sar.IBR.d2) ## This method assumes the response is known - see manual page
## This method assumes the response is known - see manual page
## This method assumes the response is known - see manual page
## Model selection table
## (Intrc) IBR PtchS family class listw df logLik AICc
## mod.sar.IBR.d1 4.067 0.09058 0.04142 () sarlm ls.d1 5 12.500 -13.9
## mod.sar.IBR.gab 4.063 0.09458 0.04081 () sarlm ls.gb 5 12.215 -13.3
## mod.sar.IBR.d2 4.055 0.10650 0.04133 () sarlm ls.d2 5 11.549 -12.0
## mod.lm.IBR 3.923 0.25520 gs(id) lm 3 8.603 -10.8
## delta weight
## mod.sar.IBR.d1 0.00 0.425
## mod.sar.IBR.gab 0.57 0.320
## mod.sar.IBR.d2 1.90 0.164
## mod.lm.IBR 3.10 0.090
## Abbreviations:
## family: gs(id) = 'gaussian(identity)'
## listw: ls.d1 = 'listw.d1', ls.d2 = 'listw.d2', ls.gb = 'listw.gab'
## Models ranked by AICc(x)
The best model (‘mod.sar.IBR.d1’) is the one with (row-standardized) inverse-distance weights (‘listw.d1’). It is only slightly better than the model with the (row-standardized) binary weights (‘listw.gab’), whereas the nonspatial model and the one with (row-standardized) inverse squared distance weights have much less support.
Let’s have a look at the best model. With the argument ‘Nagelkerke = TRUE’, we request a pseudo R-squared.
##
## Call:spatialreg::errorsarlm(formula = A ~ PatchSize + IBR, data = Dianthus.df,
## listw = listw.d1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.654613 -0.085961 0.016066 0.091687 0.389419
##
## Type: error
## Coefficients: (asymptotic standard errors)
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 4.067066 0.077809 52.2696 <2e-16
## PatchSize 0.041416 0.018554 2.2322 0.0256
## IBR 0.090578 0.094946 0.9540 0.3401
##
## Lambda: 0.22322, LR test value: 2.6442, p-value: 0.10393
## Asymptotic standard error: 0.13659
## z-value: 1.6343, p-value: 0.1022
## Wald statistic: 2.6708, p-value: 0.1022
##
## Log likelihood: 12.49972 for error model
## ML residual variance (sigma squared): 0.037595, (sigma: 0.19389)
## Nagelkerke pseudo-R-squared: 0.26613
## Number of observations: 59
## Number of parameters estimated: 5
## AIC: -14.999, (AIC for lm: -14.355)
See tutorial for ‘spmoran’ package: https://arxiv.org/ftp/arxiv/papers/1703/1703.04467.pdf
Both GLS and SAR fitted a spatially correlated error structure of a relatively simple form to the data. Gene flow could be more complex and for example, could create spatial autocorrelation structure that is not the same in all directions or in all parts of the study area. Moran Eigenvector Maps (MEM) allows a more flexible modeling of spatial structure in the data. In spatial filtering, we use MEM spatial eigenvectors to account for any spatial structure while fitting and testing the effect of our predictors.
The new package ‘spmoran’ makes this really easy. First, we create the MEM spatial eigenvectors. This implies defining neighbors and weights, but this is well hidden in the code below. The function ‘meigen’ here takes the coordinates, calculates a minimum spanning tree (so that each site has at least one neighbour), and finds the maximum distance ‘h’ from the spanning tree. It then calculates neighbor weights as exp(-dij / h).
Note: if you have many sites (> 200), the function ‘meigen_f’ may be used instead of ‘meigen’, it should even work for >1000 sites.
The function ‘esf’ then performs the spatial filtering. Here it uses stepwise selection of MEM spatial eigenvectors using an R-squared criterion (fn = “r2”).
# lm model: using truncated distance matrix (max of min spanning tree distance)
meig <- spmoran::meigen(coords=xy)## 9/59 eigen-pairs are extracted
sfd.res <- spmoran::esf( y=Dianthus.df$A, x=Dianthus.df[,c("PatchSize", "IBR")],
meig=meig, fn = "r2" )## 5/9 eigenvectors are selected
The objects created by functions ‘meigen’ and ‘esf’ contain a lot of information:
Let’s look at the table ‘b’ with regression results for the predictors first:
## Estimate SE t_value p_value
## (Intercept) 4.08678879 0.06759658 60.4585130 3.838659e-49
## PatchSize 0.03642554 0.01673612 2.1764625 3.417603e-02
## IBR 0.04687286 0.08602602 0.5448685 5.882188e-01
Again, PatchSize is statistically significant but not IBR.
Next, we look at the table ‘r’ with regression results for MEM spatial eigenvectors:
## Estimate SE t_value p_value
## sf6 0.5938646 0.1782765 3.331144 0.001613967
## sf1 0.3962440 0.1788408 2.215624 0.031207615
## sf7 -0.4065267 0.1895836 -2.144313 0.036795842
## sf3 -0.3474148 0.1760289 -1.973624 0.053854181
## sf5 -0.3290397 0.1757533 -1.872168 0.066922529
Five MEM spatial eigenvectors were important enough to be included in the model. Here they are ranked by their (absolute value of) slope coefficient, and thus by the strength of their association with the response variable. Eigenvector ‘sf6’ was by far the most important.
Note: some eigenvectors are included despite having a p-value > 0.05. This may have two reasons. First, the eigenvectors were selected without taking into account predictors X. Second, a different test was used in the stepwise eigenvector selection. The type of test can be specified with an argument ‘fn’ (see ‘?esf’ helpfile and ‘spmoran’ tutorial).
Finally, let’s look at the summary results for the fitted model:
## stat
## resid_SE 0.1728755
## adjR2 0.4374576
## logLik 24.1369653
## AIC -30.2739307
## BIC -11.5760937
Here, ‘adjR2’ is rather high (0.437), but this includes the selected MEM spatial eigenvectors!
We know already that ‘listw.d1’ fit the data well, so let’s re-run the model with our own definition of spatial weights. With the funciton ‘listw2mat’, we convert from ‘listw’ format to a full connnectivity matrix.
## Note:
## cmat is symmetrized by ( cmat + t( cmat ) ) / 2
## 27/59 eigen-pairs are extracted
sfw.res <- spmoran::esf( y=Dianthus.df$A, x=Dianthus.df[,c("PatchSize", "IBR")],
meig=meigw, fn = "r2" )## 15/27 eigenvectors are selected
## Estimate SE t_value p_value
## (Intercept) 4.07534252 0.07375191 55.2574522 3.989818e-40
## PatchSize 0.03114008 0.01772391 1.7569529 8.639209e-02
## IBR 0.05720775 0.09657088 0.5923913 5.568418e-01
## # A tibble: 15 x 4
## Estimate SE t_value p_value
## <dbl> <dbl> <dbl> <dbl>
## 1 0.666 0.171 3.90 0.000348
## 2 -0.404 0.172 -2.34 0.0240
## 3 -0.509 0.160 -3.19 0.00273
## 4 -0.424 0.171 -2.48 0.0175
## 5 0.358 0.160 2.24 0.0307
## 6 -0.314 0.167 -1.88 0.0668
## 7 0.284 0.176 1.62 0.113
## 8 -0.185 0.156 -1.18 0.243
## 9 -0.160 0.164 -0.973 0.336
## 10 -0.244 0.158 -1.54 0.131
## 11 0.260 0.163 1.60 0.117
## 12 0.358 0.184 1.94 0.0588
## 13 0.332 0.177 1.88 0.0679
## 14 -0.237 0.162 -1.47 0.151
## 15 0.215 0.159 1.35 0.185
## stat
## resid_SE 0.1520845
## adjR2 0.5646300
## logLik 38.1354212
## AIC -38.2708424
## BIC 1.2023691
Note: the messages tell us that ‘cmat’ has been made symmetric before analysis, that 27 out of 59 MEM spatial eigenvector (and their eigenvalues, hence ‘pairs’) were retained initially and subjected to stepwise selection, which then returned 15 statistically significant MEM eigenvectors that were included in the regression model with the predictor variables X (PatchSize and IBR).
Questions:
So far, we have treated the MEM spatial eigenvectors as a black box. What kind of patterns do they represent?
First, we plot all the selected (significant) eigenvectors. A convenient way to do so is converting to a SpatialPointsDataFrame and then use the function ‘spplot’.
Here we need tweak the column names of the eigenvectors, which are called “X1”, “X2” etc., to match the row names of table ‘sf.res$r’, which uses “sf1”, “sf2” etc.
SF <- data.frame(xy, sf=meigw$sf)
names(SF) <- gsub("[.]", "", names(SF))
sp::coordinates(SF) <- ~ x + y
sp::spplot(SF, row.names(sfw.res$r), colorkey = TRUE)The panel starts at the bottom left, so that the most important spatial eigenvector (sf6) is plotted at the bottom left, the second most important (sf22) second from left, etc.
The smallest numbers are patterns with the largest spatial scale (sf1), which here shows a gradient from East to West. The most important eigenvector (sf6) shows a finer-scale pattern with the highest values (yellow) in the center, lowest values East and West, and intermediate value North and South.
However, these patterns individually are not meaningful. More importantly, we can plot the total spatial component in the response as a weighted sum of these component patterns, where the weights correspond to the regression coefficients in table ‘sf.res$r’. Here we create a panel with two plots, the modeled spatial components ‘sf’ on the left and the response ‘A’ on the right (the mean has been removed to make values comparable).
SF@data$sf <- sfw.res$sf
SF@data$A <- scale(Dianthus.df$A, scale = FALSE)
sp::spplot(SF, c("sf", "A"), colorkey = TRUE)Obviously, a big part of the variation in allelic richness is already captured by ‘sf’. In essence the model then tries to explain the difference between these two sets of values by the predictors “PatchSize” and “IBR”.
Let’s quantify the correlation of this spatial component with allelic richness, and compare the correlation between the two models:
## sfd sfw
## [1,] 0.6252594 0.7720016
With the default method (defining neighbors based on a distance cut-off), the spatial component modeled by the significant MEM spatial eigenvectors showed a correlation of 0.625 with the response variable. Using a Gabriel graph with inverse distance weights increased this correlation to 0.772.
This means that the spatial eigenvectors derived from the Gabriel graph were more effective at capturing the spatial variation in allelic richness than the default method. This spatial component is then controlled for when assessing the relationship between allelic richness and the predictors (PatchSize and IBR).
The previous model selected 15 MEM spatial eigenvectors, and thus fitted 15 additional models. Just like the random effects for family and population in Week 6 lab, we can save a few parameters here by fitting the set of MEM eigenvectors as a random effect. This is done by the function ‘resf’.
sfr.res <- spmoran::resf( y=Dianthus.df$A, x=Dianthus.df[,c("PatchSize", "IBR")],
meig = meigw, method = "reml" )
sfr.res$b## Estimate SE t_value p_value
## (Intercept) 4.05889381 0.06856506 59.197696 0.00000000
## PatchSize 0.04215982 0.01675185 2.516726 0.01531481
## IBR 0.10077818 0.08727232 1.154755 0.25402637
## # A tibble: 27 x 3
## Estimate SE t_value
## <dbl> <dbl> <dbl>
## 1 0.164 0.121 1.35
## 2 -0.0408 0.121 -0.338
## 3 0.0324 0.120 0.271
## 4 -0.0999 0.122 -0.820
## 5 0.118 0.126 0.938
## 6 0.243 0.125 1.95
## 7 0.0281 0.126 0.224
## 8 0.0691 0.126 0.550
## 9 -0.174 0.124 -1.40
## 10 -0.0201 0.126 -0.160
## # … with 17 more rows
## stat
## resid_SE 0.1636501
## adjR2(cond) 0.4571174
## logLik 4.7082457
## AIC 2.5835085
## BIC 15.0487332
## Estimate
## spcomp_SE 0.1739337
## spcomp_Moran.I/max(Moran.I) 0.7587574
As in Week 6 lab, the conditional R-squared is the variance explained by the fixed effects (PatchSize and IBR) and the random effects (significant MEM spatial eigenvectors) together. It is adjusted for the number of effects that were estimated.
Note: we can’t compare AIC with the previous models, as the model was fitted with ‘reml’.
We get an additional output ‘sfr.res$s’ with two variance parameters:
See: https://arxiv.org/ftp/arxiv/papers/1703/1703.04467.pdf
Now comes the coolest part!
So far, we have fitted the same model for all sites. Geographically weighted regression (GWR) would allow relaxing this. Spatial filtering with MEM can be used to accomplish the same goal, and the ‘spmoran’ tutorial calls this a ‘Spatially Varying Coefficients’ model (SVC). The main advantage is that we can visualize how the slope parameter estimates, and their p-values, vary across the study area! This is a great exploratory tool that can help us better understand what is going on.
We fit the model with ‘resf_vc’.
rv_res <- spmoran::resf_vc( y=Dianthus.df$A,
x = Dianthus.df[,c("PatchSize", "IBR")],
xconst = NULL, meig = meigw, method = "reml", allsvc=TRUE)## [1] "------- Iteration 1 -------"
## [1] "1/3"
## [1] "2/3"
## [1] "3/3"
## [1] "rlogLik: 6.898"
## [1] "------- Iteration 2 -------"
## [1] "1/3"
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Instead of one slope estimate for each predictor, we now get a different estimate for each combination of parameter and site (sounds like overfitting?). Here’s a summary of the distbiution of these estimates.
## (Intercept) PatchSize IBR
## Min. :4.121 Min. :0.0007725 Min. :-0.812764
## 1st Qu.:4.121 1st Qu.:0.0280929 1st Qu.:-0.077299
## Median :4.121 Median :0.0453303 Median : 0.057376
## Mean :4.121 Mean :0.0457974 Mean :-0.001217
## 3rd Qu.:4.121 3rd Qu.:0.0639628 3rd Qu.: 0.138549
## Max. :4.121 Max. :0.0979951 Max. : 0.336830
The slope estimate for PatchSize varied between 0 and 0.098, with a mean of 0.046. The slope estimate for the ‘IBR’ term varied between -0.81 and 0.34, with a mean very close to 0! That is an astounding range of variation. Keep in mind that we really expect a positive relationship, there is no biological explanation for a ngeative relationship.
Here is a similar summary of the p-values:
## (Intercept) PatchSize IBR
## Min. :0 Min. :0.0000077 Min. :0.0000091
## 1st Qu.:0 1st Qu.:0.0149433 1st Qu.:0.0780294
## Median :0 Median :0.1058639 Median :0.2391374
## Mean :0 Mean :0.2328291 Mean :0.3197722
## 3rd Qu.:0 3rd Qu.:0.3833774 3rd Qu.:0.5319126
## Max. :0 Max. :0.9839952 Max. :0.9058305
For both variables, most sites do not show a significant effect (i.e., only few sites show a p-value < 0.05).
We could print these results by site (type ‘rv_res\(b_vc' or 'rv_res\)p_vc’). Even better, we can plot them in space. We start with combining the data (‘Dianthus.df’) and the results into one data frame ‘Results’. By specifying ‘b=rv_res\(b_vc' and "p=rv_res\)p_vc’, R will create column names that start with ‘b’ or ‘p’, respectively.
## [1] "A" "IBD" "IBR" "PatchSize"
## [5] "Longitude" "Latitude" "x" "y"
## [9] "b..Intercept." "b.PatchSize" "b.IBR" "p..Intercept."
## [13] "p.PatchSize" "p.IBR"
Let’s start with PatchSize. Here, we first plot PatchSize in space, with symbol size as a function of patch size. In a second plot, we color sites by statistical significance and the size of the symbols represents the parameter estimate of the regression slope coefficient for Patch Size. The layer ‘coord_fixed’ keeps controls the aspect ratio between x- and y-axes.
require(ggplot2)
ggplot(as.data.frame(Result), aes(x, y, size=PatchSize)) +
geom_point(color="darkblue") + coord_fixed()
ggplot(as.data.frame(Result), aes(x, y, col=p.PatchSize < 0.05, size=b.PatchSize)) +
geom_point() + coord_fixed()Let’s do the same for ‘IBR’:
require(ggplot2)
ggplot(as.data.frame(Result), aes(x, y, size=IBR)) +
geom_point(color="darkgreen") + coord_fixed()
ggplot(as.data.frame(Result), aes(x, y, col=p.IBR < 0.05, size=b.IBR)) +
geom_point() + coord_fixed()Keep in mind that ‘IBR’ and ‘PatchSize’ showed a strong correlation. The parameter estimates could therefore depend quite a bit on the other variables. To help with the interpretation, let’s repeat the last analysis just with ‘IBR’, without ‘PatchSize’.
rv_res <- spmoran::resf_vc( y=Dianthus.df$A,
x = Dianthus.df[,c("IBR")],
xconst = NULL, meig = meigw, method = "reml", allsvc=TRUE)## [1] "------- Iteration 1 -------"
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## (Intercept) V1
## Min. :3.811 Min. :-0.2388
## 1st Qu.:3.915 1st Qu.: 0.1544
## Median :3.938 Median : 0.2589
## Mean :3.939 Mean : 0.2111
## 3rd Qu.:3.966 3rd Qu.: 0.3203
## Max. :4.045 Max. : 0.4317
Now the range of slope estimates is smaller, most sites have a positive estimate, and the mean is approx. 0.21.
## (Intercept) V1
## Min. :0 Min. :0.000209
## 1st Qu.:0 1st Qu.:0.004605
## Median :0 Median :0.032435
## Mean :0 Mean :0.170820
## 3rd Qu.:0 3rd Qu.:0.223984
## Max. :0 Max. :0.983408
Also, a larger proportion of sites nows has p-values < 0.05.
Let’s plot the results onto a gray-scale, stamen ‘terrain map to facilitate interpretation. Note: here the zoom level ’zoom = 12’ covers the entire study area, whereas the default value would actually cut off a large number of sites. We use the argument ‘force=TRUE’ to force the map to be downloaded again (otherwise the argument ‘color=“bw”’ may not have an effect if we already downloaded the terrain map in color).
## [1] "A" "IBD" "IBR" "PatchSize"
## [5] "Longitude" "Latitude" "x" "y"
## [9] "b..Intercept." "b.V1" "p..Intercept." "p.V1"
## [13] "resid"
ggmap::ggmap(get_stamenmap(bbox, maptype = "terrain", color="bw", force=TRUE, zoom=12)) +
geom_point(aes(x = Longitude, y = Latitude, col=p.V1 < 0.05, size=b.V1), data = Result) ## Source : http://tile.stamen.com/terrain/12/2172/1405.png
## Source : http://tile.stamen.com/terrain/12/2173/1405.png
## Source : http://tile.stamen.com/terrain/12/2174/1405.png
## Source : http://tile.stamen.com/terrain/12/2172/1406.png
## Source : http://tile.stamen.com/terrain/12/2173/1406.png
## Source : http://tile.stamen.com/terrain/12/2174/1406.png
## Source : http://tile.stamen.com/terrain/12/2172/1407.png
## Source : http://tile.stamen.com/terrain/12/2173/1407.png
## Source : http://tile.stamen.com/terrain/12/2174/1407.png
## Source : http://tile.stamen.com/terrain/12/2172/1408.png
## Source : http://tile.stamen.com/terrain/12/2173/1408.png
## Source : http://tile.stamen.com/terrain/12/2174/1408.png
This is a very different map of results!